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Benford's Law

Benford's Law - also known as the Newcombā€“Benford law, was originally stated by astronomer Simon Newcomb in 1881, and was subsequently re-stated by physicist Frank Benford in 1938.

Benford originally called his thesis 'The Law of Anomalous Numbers' (ref.Proceedings of the American Philosophical Society, Vol. 78, No. 4, pp.551-572)

It asserts that in many large sets of real-world data, the leading digit of a data entry is much more likely to be lower (e.g. 1) than higher (e.g. 9)

Simon Newcomb noticed the phenomenon in tables of logarithms which he was using in his work. Specifically, he noted that some pages of a library's logarithmic-table books were much more worn-by-use than were others.

It was later found that the law strongly applies in fields as diverse as stock market prices, house numbers in streets, the half-lives of unstable radioactive materials, social-media follower numbers, and many other areas. In these cases, numbers with the leading-digit 1 tend to occur roughly 30% of the time, while leading-digit 9 occurs in less than 5% of cases.

Note that in trulyrandomplugin-autotooltip__plain plugin-autotooltip_bigRandom numbers


"We can never decide for sure that a number is random, but what we can do is apply an increasing number of tests and treat our sequence of numbers as innocent until proved guilty."

Source Colva Roney-Dougal, Senior Lecturer in Pure Mathematics at the
large sets of numbers, all digits occur with the same frequency. In other words, one of the hints given by the law is that some sets of real-world data which might be assumed to be random could in reality be substantially non-random.

It should also be noted that many of the often-cited examples of the law in action - e.g. lists of the heights of buildings - rely completely on the use of a particular unit of measurement. Some of which were originally developed for fairly arbitrary psycho-social reasons.

As an example, a list of the average annual temperatures in major cites of the world - measured in Centigrade - would feature many leading digits 1, 2 and 3 but very few 4s, and no 5s, 6s and 7s etc. But, if the unit of measurement was Kelvin, the law would no longer apply. This may say more about human preferences for small(ish) numbers than it does about hard-wired mathematics.

There have been many attempts at explaining the law - but none has been universally accepted by the mathematics community as a rigorous explanation for all cases where the law has been found to apply. (Example study A statistical derivation of the significant-digit law ,Stat. Sci., 10 (4) pp. 354-363) Further technical info Wolfram Mathworld

Note : In 2023, a research team from Peking University, Beijing, claim to have a proof of the law - though the global mathematical community are yet to formally accept it. See :Fundamental Research currently in-press, April 2023.

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